Theorem List for Intuitionistic Logic Explorer - 11001-11100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | minmax 11001 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
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inf |
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Theorem | mincl 11002 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
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inf |
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Theorem | min1inf 11003 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
|
inf |
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Theorem | min2inf 11004 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
|
inf |
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Theorem | lemininf 11005 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by NM, 3-Aug-2007.)
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inf
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Theorem | ltmininf 11006 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
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inf |
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Theorem | minabs 11007 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
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inf
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Theorem | minclpr 11008 |
The minimum of two real numbers is one of those numbers if and only if
dichotomy (
) holds. For example, this
can be
combined with zletric 9098 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 23-May-2023.)
|
inf
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Theorem | rpmincl 11009 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
inf |
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Theorem | bdtrilem 11010 |
Lemma for bdtri 11011. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
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Theorem | bdtri 11011 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
inf
inf inf |
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Theorem | mul0inf 11012 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 10834 and mulap0bd 8418 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
inf |
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4.7.7 The maximum of two extended
reals
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Theorem | xrmaxleim 11013 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
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Theorem | xrmaxiflemcl 11014 |
Lemma for xrmaxif 11020. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
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Theorem | xrmaxifle 11015 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxiflemab 11016 |
Lemma for xrmaxif 11020. A variation of xrmaxleim 11013- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxiflemlub 11017 |
Lemma for xrmaxif 11020. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
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Theorem | xrmaxiflemcom 11018 |
Lemma for xrmaxif 11020. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
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Theorem | xrmaxiflemval 11019* |
Lemma for xrmaxif 11020. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
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Theorem | xrmaxif 11020 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
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Theorem | xrmaxcl 11021 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
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Theorem | xrmax1sup 11022 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
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Theorem | xrmax2sup 11023 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
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Theorem | xrmaxrecl 11024 |
The maximum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
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Theorem | xrmaxleastlt 11025 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
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Theorem | xrltmaxsup 11026 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
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Theorem | xrmaxltsup 11027 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
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Theorem | xrmaxlesup 11028 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 10-May-2023.)
|
|
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Theorem | xrmaxaddlem 11029 |
Lemma for xrmaxadd 11030. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
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Theorem | xrmaxadd 11030 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
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4.7.8 The minimum of two extended
reals
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Theorem | xrnegiso 11031 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
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Theorem | infxrnegsupex 11032* |
The infimum of a set of extended reals is the negative of the
supremum of the negatives of its elements. (Contributed by Jim Kingdon,
2-May-2023.)
|
inf
|
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Theorem | xrnegcon1d 11033 |
Contraposition law for extended real unary minus. (Contributed by Jim
Kingdon, 2-May-2023.)
|
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Theorem | xrminmax 11034 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
2-May-2023.)
|
inf
|
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Theorem | xrmincl 11035 |
The minumum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 3-May-2023.)
|
inf |
|
Theorem | xrmin1inf 11036 |
The minimum of two extended reals is less than or equal to the first.
(Contributed by Jim Kingdon, 3-May-2023.)
|
inf |
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Theorem | xrmin2inf 11037 |
The minimum of two extended reals is less than or equal to the second.
(Contributed by Jim Kingdon, 3-May-2023.)
|
inf |
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Theorem | xrmineqinf 11038 |
The minimum of two extended reals is equal to the second if the first is
bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim
Kingdon, 3-May-2023.)
|
inf
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Theorem | xrltmininf 11039 |
Two ways of saying an extended real is less than the minimum of two
others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
3-May-2023.)
|
inf |
|
Theorem | xrlemininf 11040 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 4-May-2023.)
|
inf |
|
Theorem | xrminltinf 11041 |
Two ways of saying an extended real is greater than the minimum of two
others. (Contributed by Jim Kingdon, 19-May-2023.)
|
inf
|
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Theorem | xrminrecl 11042 |
The minimum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|
inf inf |
|
Theorem | xrminrpcl 11043 |
The minimum of two positive reals is a positive real. (Contributed by Jim
Kingdon, 4-May-2023.)
|
inf |
|
Theorem | xrminadd 11044 |
Distributing addition over minimum. (Contributed by Jim Kingdon,
10-May-2023.)
|
inf inf |
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Theorem | xrbdtri 11045 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
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inf
inf inf
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Theorem | iooinsup 11046 |
Intersection of two open intervals of extended reals. (Contributed by
NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|
inf |
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4.8 Elementary limits and
convergence
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4.8.1 Limits
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Syntax | cli 11047 |
Extend class notation with convergence relation for limits.
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Definition | df-clim 11048* |
Define the limit relation for complex number sequences. See clim 11050
for
its relational expression. (Contributed by NM, 28-Aug-2005.)
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Theorem | climrel 11049 |
The limit relation is a relation. (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | clim 11050* |
Express the predicate: The limit of complex number sequence is
, or converges to . This means that for any
real
, no matter how
small, there always exists an integer such
that the absolute difference of any later complex number in the sequence
and the limit is less than . (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | climcl 11051 |
Closure of the limit of a sequence of complex numbers. (Contributed by
NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
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Theorem | clim2 11052* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 11050. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
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Theorem | clim2c 11053* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | clim0 11054* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | clim0c 11055* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climi 11056* |
Convergence of a sequence of complex numbers. (Contributed by NM,
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climi2 11057* |
Convergence of a sequence of complex numbers. (Contributed by NM,
11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climi0 11058* |
Convergence of a sequence of complex numbers to zero. (Contributed by
NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climconst 11059* |
An (eventually) constant sequence converges to its value. (Contributed
by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climconst2 11060 |
A constant sequence converges to its value. (Contributed by NM,
6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climz 11061 |
The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.)
(Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climuni 11062 |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro,
31-Jan-2014.)
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Theorem | fclim 11063 |
The limit relation is function-like, and with range the complex numbers.
(Contributed by Mario Carneiro, 31-Jan-2014.)
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Theorem | climdm 11064 |
Two ways to express that a function has a limit. (The expression
is sometimes useful as a shorthand for "the unique limit
of the function "). (Contributed by Mario Carneiro,
18-Mar-2014.)
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Theorem | climeu 11065* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 25-Dec-2005.)
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Theorem | climreu 11066* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by NM, 25-Dec-2005.)
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Theorem | climmo 11067* |
An infinite sequence of complex numbers converges to at most one limit.
(Contributed by Mario Carneiro, 13-Jul-2013.)
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Theorem | climeq 11068* |
Two functions that are eventually equal to one another have the same
limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario
Carneiro, 31-Jan-2014.)
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Theorem | climmpt 11069* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
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Theorem | 2clim 11070* |
If two sequences converge to each other, they converge to the same
limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario
Carneiro, 31-Jan-2014.)
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Theorem | climshftlemg 11071 |
A shifted function converges if the original function converges.
(Contributed by Mario Carneiro, 5-Nov-2013.)
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Theorem | climres 11072 |
A function restricted to upper integers converges iff the original
function converges. (Contributed by Mario Carneiro, 13-Jul-2013.)
(Revised by Mario Carneiro, 31-Jan-2014.)
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Theorem | climshft 11073 |
A shifted function converges iff the original function converges.
(Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro,
31-Jan-2014.)
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Theorem | serclim0 11074 |
The zero series converges to zero. (Contributed by Paul Chapman,
9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
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Theorem | climshft2 11075* |
A shifted function converges iff the original function converges.
(Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario
Carneiro, 6-Feb-2014.)
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Theorem | climabs0 11076* |
Convergence to zero of the absolute value is equivalent to convergence
to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro,
31-Jan-2014.)
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Theorem | climcn1 11077* |
Image of a limit under a continuous map. (Contributed by Mario
Carneiro, 31-Jan-2014.)
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Theorem | climcn2 11078* |
Image of a limit under a continuous map, two-arg version. (Contributed
by Mario Carneiro, 31-Jan-2014.)
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Theorem | addcn2 11079* |
Complex number addition is a continuous function. Part of Proposition
14-4.16 of [Gleason] p. 243. (We write
out the definition directly
because df-cn and df-cncf are not yet available to us. See addcncntop 12721
for the abbreviated version.) (Contributed by Mario Carneiro,
31-Jan-2014.)
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Theorem | subcn2 11080* |
Complex number subtraction is a continuous function. Part of
Proposition 14-4.16 of [Gleason] p. 243.
(Contributed by Mario
Carneiro, 31-Jan-2014.)
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Theorem | mulcn2 11081* |
Complex number multiplication is a continuous function. Part of
Proposition 14-4.16 of [Gleason] p. 243.
(Contributed by Mario
Carneiro, 31-Jan-2014.)
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Theorem | reccn2ap 11082* |
The reciprocal function is continuous. The class is just for
convenience in writing the proof and typically would be passed in as an
instance of eqid 2139. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
inf #
#
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Theorem | cn1lem 11083* |
A sufficient condition for a function to be continuous. (Contributed by
Mario Carneiro, 9-Feb-2014.)
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Theorem | abscn2 11084* |
The absolute value function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
|
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Theorem | cjcn2 11085* |
The complex conjugate function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
|
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Theorem | recn2 11086* |
The real part function is continuous. (Contributed by Mario Carneiro,
9-Feb-2014.)
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Theorem | imcn2 11087* |
The imaginary part function is continuous. (Contributed by Mario
Carneiro, 9-Feb-2014.)
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Theorem | climcn1lem 11088* |
The limit of a continuous function, theorem form. (Contributed by
Mario Carneiro, 9-Feb-2014.)
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Theorem | climabs 11089* |
Limit of the absolute value of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
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Theorem | climcj 11090* |
Limit of the complex conjugate of a sequence. Proposition 12-2.4(c)
of [Gleason] p. 172. (Contributed by
NM, 7-Jun-2006.) (Revised by
Mario Carneiro, 9-Feb-2014.)
|
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Theorem | climre 11091* |
Limit of the real part of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
|
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Theorem | climim 11092* |
Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of
[Gleason] p. 172. (Contributed by NM,
7-Jun-2006.) (Revised by Mario
Carneiro, 9-Feb-2014.)
|
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Theorem | climrecl 11093* |
The limit of a convergent real sequence is real. Corollary 12-2.5 of
[Gleason] p. 172. (Contributed by NM,
10-Sep-2005.)
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Theorem | climge0 11094* |
A nonnegative sequence converges to a nonnegative number. (Contributed
by NM, 11-Sep-2005.)
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Theorem | climadd 11095* |
Limit of the sum of two converging sequences. Proposition 12-2.1(a)
of [Gleason] p. 168. (Contributed
by NM, 24-Sep-2005.) (Proof
shortened by Mario Carneiro, 31-Jan-2014.)
|
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Theorem | climmul 11096* |
Limit of the product of two converging sequences. Proposition
12-2.1(c) of [Gleason] p. 168.
(Contributed by NM, 27-Dec-2005.)
(Proof shortened by Mario Carneiro, 1-Feb-2014.)
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Theorem | climsub 11097* |
Limit of the difference of two converging sequences. Proposition
12-2.1(b) of [Gleason] p. 168.
(Contributed by NM, 4-Aug-2007.)
(Proof shortened by Mario Carneiro, 1-Feb-2014.)
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Theorem | climaddc1 11098* |
Limit of a constant
added to each term of a sequence.
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
3-Feb-2014.)
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Theorem | climaddc2 11099* |
Limit of a constant
added to each term of a sequence.
(Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro,
3-Feb-2014.)
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Theorem | climmulc2 11100* |
Limit of a sequence multiplied by a constant . Corollary
12-2.2 of [Gleason] p. 171.
(Contributed by NM, 24-Sep-2005.)
(Revised by Mario Carneiro, 3-Feb-2014.)
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